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Full text of "Mathematical And Physical Papers - Iii"

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the mass, except so far as it may be raised or lowered by sudden condensation or rarefaction in the case of an elastic fluid. The general equations contain the differential coefficients of the quantity fji with respect to x, y, and z\ but the equations of the form (1) are in their present shape even more general than is required for the purposes of the present paper.
These equations agree in the main with those which had been previously obtained, on different principles, by Navier, by Poisson, and by M. de Saint-Yenant, as I have elsewhere observed*. The differences depend only on the coefficient of the last term, and this term vanishes in the case of an incompressible fluid, to which Navier had confined his investigations.
jsuch as (1) in their present shape are rather,,^
complicated, but in applying them to the case of a pendulum they may be a good deal simplified without the neglect of any quantities which it would be important to retain. In the first place the motion is supposed very small, on which account it will be allowable to neglect the terms which involve the square of the velocity. In the second place, the nature of the motion that we have got to deal with is such that the compressibility of the fluid has very little influence on the result, so that we may treat the fluid as incompressible, and consequently omit the last terms in the equations. Lastly, the forces X, Y, Z&TQ in the present case the components of the force of gravity, and if we write
for jp, we may omit the terms X, Y, Z.
If / be measured vertically downwards from a horizontal plane drawn in the neighbourhood of the pendulum, and if g be the force of gravity, $(Xdx+Ydy + Zdz)=g2f, the arbitrary constant, or arbitrary function of the time if it should be found necessary to suppose it to be such, being included in IT. The part of the whole force acting on the pendulum which depends on the terms II -f gpzf is simply a force equal to the weight of the fluid displaced, and acting vertically upwards through the centre of gravity of the volume.
* Eeport on recent researches in Hydrodynamics.   Eeport of the British Association for 1846, p. 16.    [Ante, Vol. i. p. 182.]